# American Institue of Mathematical Sciences

2018, 10(1): 1-41. doi: 10.3934/jgm.2018001

## Lagrange-d'Alembert-Poincaré Equations by Several Stages

 Departamento de Matemática, Universidad Nacional del Sur, Av. Alem 1253,8000 Bahía Blanca, Argentina

Received  June 2014 Revised  June 2017 Published  December 2017

The aim of this paper is to write explicit expression in terms of a given principal connection of the Lagrange-d'Alembert-Poincaré equations by several stages. This is obtained by using a reduced Lagrange-d'Alembert's Principle by several stages, extending methods known for the case of one stage in the previous literature. The case of Euler's disk is described as an illustrative example.

Citation: Hernán Cendra, Viviana A. Díaz. Lagrange-d'Alembert-Poincaré Equations by Several Stages. Journal of Geometric Mechanics, 2018, 10 (1) : 1-41. doi: 10.3934/jgm.2018001
##### References:
 [1] R. Abraham and J. E. Marsden, Foundation of Mechanics, Addison Wesley, second edition, 1978. [2] V. I. Arnold, Mathematical Methods of Classical Mechanics, volume60 of Graduate Texts in Mathematics. Springer Verlag, second edition, 1989. [3] L. Bates, H. Graumann, C. MacDonnell, Examples of Gauge conservation laws in nonholonomic systems, Rep. Math. Phys., 37 (1996), 295-308. doi: 10.1016/0034-4877(96)84069-9. [4] L. Bates, J. Sniatycki, Nonholonomic reduction, Reports on Math. Phys., 32 (1993), 99-115. doi: 10.1016/0034-4877(93)90073-N. [5] A. M. Bloch, Nonholonomic Mechanics and Control, volume24 of Interdisciplinary Applied Mathematics. Springer Verlag, 2003. [6] A. M. Bloch, J. E. Krishnaprasad, J. E. Marsden, R. Murray, Nonholonomic mechanical systems with symmetry, Arch. Rat. Mech. An., 136 (1996), 21-99. doi: 10.1007/BF02199365. [7] F. Cantrijn, M. deLeón, J. C. Marrero, D. Martín de Diego, Reduction of constrained systems with symmetries, J. Math. Phys., 40 (1999), 795-820. doi: 10.1063/1.532686. [8] J. J. Cariñena, M. F. Rañada, Lagrangian systems with constraints: A geometric approach to the method of Lagrange multipliers, J. Phys. A: Math. Gen., 26 (1993), 1335-1351. doi: 10.1088/0305-4470/26/6/016. [9] H. Cendra, V. A. Díaz, The Lagrange-d'Alembert-Poincaré equations and integrability for the Euler's disk, Regular and Chaotic Dynamics, 12 (2007), 56-67. doi: 10.1134/S1560354707010054. [10] H. Cendra, J. E. Marsden, T. S. Ratiu, Geometric mechanics, Lagrangian reduction and nonholonomic systems, Mathematics Unlimited and Beyond, Springer, (2001), 221-273. [11] H. Cendra, J. E. Marsden and T. S. Ratiu, Lagrangian reduction by stages, Memoirs of the AMS, 152 (2001), x+108 pp. [12] H. Cendra, J. E. Marsden, T. S. Ratiu and H. Yoshimura, Dirac-Weinstein reduction of Dirac anchored vector bundles, 2009, preprint. [13] J. Cortés, M. l deLeón, J.C. Marrero, E. Martínez, Nonholonomic Lagrangian systems on Lie algebroids, Discrete Contin. Dyn. Syst., 24 (2009), 213-271. doi: 10.3934/dcds.2009.24.213. [14] M. deLeón, D. Martínde Diego, On the geometry of non-holonomic Lagrangian systems, J. Math. Phys., 37 (1996), 3389-3414. doi: 10.1063/1.531571. [15] M. deLeón, J.C. Marrero, E. Martínez, Lagrangian submanifolds and dynamics on Lie algebroids, J. Phys. A, 38 (2005), R241-R308. [16] K. Ehlers, J. Koiller, R. Montgomery, P. M. Ríos, Nonholonomic systems via moving frames: Cartan equivalence and Chaplygin Hamiltonization, The Breath of symplectic and Poisson geometry. Prog. Math., 232 (2005), 75-120. [17] R. L. Fernandes, Lie algebroids, holonomy and characteristic classes, Adv. Math., 170 (2002), 119-179. [18] D.D. Holm, J.E. Marsden, T.S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. Math., 137 (1998), 1-81. [19] W. S. Koon, J. E. Marsden, Optimal control for holonomic and nonholonomic mechanical systems with syummetry and Lagrangian reduction, SIAM J. Control Optim., 35 (1997), 901-929. [20] Ch.M. Marle, Reduction of constrained mechanical systems and stability of relative equilibria, Commun. Math. Phys., 174 (1995), 295-318. [21] Ch.M. Marle, Various approaches to conservative and nonconservative nonholonomic systems, Rep. Math. Phys., 42 (1998), 211-229. [22] J. E. Marsden, G. Misiolek, J. P. Ortega, M. Perlmutter and T. S. Ratiu, Hamiltonian Reduction by Stages Number 1913 in Hamiltonian Reduction by Stages. Springer, 2007. [23] J. E. Marsden and T. Ratiu, Introduction to Mechanics and Symmetry, volume17. Springer-Verlag, New York, 1994. Second edition, 1999. [24] T. Mestdag, Lagrangian reduction by stages for non-holonomic systems in a Lie algebroid framework, J. Phys. A, 38 (2005), 10157-10179. [25] J. I. Neimark and N. A. Fufaev, Dynamics of Nonholonomic Systems, Translations of the American Mathematical Society, Providence, Rhode Island, 1972. [26] A. M. Vershik, Classical and Non-Classical Dynamics with Constraints, volume 1108 of Global Analysis-Studies and Applications I. Lecture Notes in Mathematics. Springer, 2002. [27] A. M. Vershik, L. D. Faddeev, Differential geometry and Lagrangian mechanics with constraints, Sov. Phys. Dokl., 17 (1972), 34-36.

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##### References:
 [1] R. Abraham and J. E. Marsden, Foundation of Mechanics, Addison Wesley, second edition, 1978. [2] V. I. Arnold, Mathematical Methods of Classical Mechanics, volume60 of Graduate Texts in Mathematics. Springer Verlag, second edition, 1989. [3] L. Bates, H. Graumann, C. MacDonnell, Examples of Gauge conservation laws in nonholonomic systems, Rep. Math. Phys., 37 (1996), 295-308. doi: 10.1016/0034-4877(96)84069-9. [4] L. Bates, J. Sniatycki, Nonholonomic reduction, Reports on Math. Phys., 32 (1993), 99-115. doi: 10.1016/0034-4877(93)90073-N. [5] A. M. Bloch, Nonholonomic Mechanics and Control, volume24 of Interdisciplinary Applied Mathematics. Springer Verlag, 2003. [6] A. M. Bloch, J. E. Krishnaprasad, J. E. Marsden, R. Murray, Nonholonomic mechanical systems with symmetry, Arch. Rat. Mech. An., 136 (1996), 21-99. doi: 10.1007/BF02199365. [7] F. Cantrijn, M. deLeón, J. C. Marrero, D. Martín de Diego, Reduction of constrained systems with symmetries, J. Math. Phys., 40 (1999), 795-820. doi: 10.1063/1.532686. [8] J. J. Cariñena, M. F. Rañada, Lagrangian systems with constraints: A geometric approach to the method of Lagrange multipliers, J. Phys. A: Math. Gen., 26 (1993), 1335-1351. doi: 10.1088/0305-4470/26/6/016. [9] H. Cendra, V. A. Díaz, The Lagrange-d'Alembert-Poincaré equations and integrability for the Euler's disk, Regular and Chaotic Dynamics, 12 (2007), 56-67. doi: 10.1134/S1560354707010054. [10] H. Cendra, J. E. Marsden, T. S. Ratiu, Geometric mechanics, Lagrangian reduction and nonholonomic systems, Mathematics Unlimited and Beyond, Springer, (2001), 221-273. [11] H. Cendra, J. E. Marsden and T. S. Ratiu, Lagrangian reduction by stages, Memoirs of the AMS, 152 (2001), x+108 pp. [12] H. Cendra, J. E. Marsden, T. S. Ratiu and H. Yoshimura, Dirac-Weinstein reduction of Dirac anchored vector bundles, 2009, preprint. [13] J. Cortés, M. l deLeón, J.C. Marrero, E. Martínez, Nonholonomic Lagrangian systems on Lie algebroids, Discrete Contin. Dyn. Syst., 24 (2009), 213-271. doi: 10.3934/dcds.2009.24.213. [14] M. deLeón, D. Martínde Diego, On the geometry of non-holonomic Lagrangian systems, J. Math. Phys., 37 (1996), 3389-3414. doi: 10.1063/1.531571. [15] M. deLeón, J.C. Marrero, E. Martínez, Lagrangian submanifolds and dynamics on Lie algebroids, J. Phys. A, 38 (2005), R241-R308. [16] K. Ehlers, J. Koiller, R. Montgomery, P. M. Ríos, Nonholonomic systems via moving frames: Cartan equivalence and Chaplygin Hamiltonization, The Breath of symplectic and Poisson geometry. Prog. Math., 232 (2005), 75-120. [17] R. L. Fernandes, Lie algebroids, holonomy and characteristic classes, Adv. Math., 170 (2002), 119-179. [18] D.D. Holm, J.E. Marsden, T.S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. Math., 137 (1998), 1-81. [19] W. S. Koon, J. E. Marsden, Optimal control for holonomic and nonholonomic mechanical systems with syummetry and Lagrangian reduction, SIAM J. Control Optim., 35 (1997), 901-929. [20] Ch.M. Marle, Reduction of constrained mechanical systems and stability of relative equilibria, Commun. Math. Phys., 174 (1995), 295-318. [21] Ch.M. Marle, Various approaches to conservative and nonconservative nonholonomic systems, Rep. Math. Phys., 42 (1998), 211-229. [22] J. E. Marsden, G. Misiolek, J. P. Ortega, M. Perlmutter and T. S. Ratiu, Hamiltonian Reduction by Stages Number 1913 in Hamiltonian Reduction by Stages. Springer, 2007. [23] J. E. Marsden and T. Ratiu, Introduction to Mechanics and Symmetry, volume17. Springer-Verlag, New York, 1994. Second edition, 1999. [24] T. Mestdag, Lagrangian reduction by stages for non-holonomic systems in a Lie algebroid framework, J. Phys. A, 38 (2005), 10157-10179. [25] J. I. Neimark and N. A. Fufaev, Dynamics of Nonholonomic Systems, Translations of the American Mathematical Society, Providence, Rhode Island, 1972. [26] A. M. Vershik, Classical and Non-Classical Dynamics with Constraints, volume 1108 of Global Analysis-Studies and Applications I. Lecture Notes in Mathematics. Springer, 2002. [27] A. M. Vershik, L. D. Faddeev, Differential geometry and Lagrangian mechanics with constraints, Sov. Phys. Dokl., 17 (1972), 34-36.
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